Question
Consider a square lawn that measures 20 ft×20 ft
. Along one side of the lawn, there is a semicircular flower garden with a diameter of 10 ft
.
A square with a semicircle inside the square with the diameter of the circle on top of the left side of the square. The square’s side is labeled 20 feet and the semicircle’s diameter is labeled 10 feet.
A bird drops a seed randomly on the lawn. What is the probability that the seed lands outside the semicircular flower garden?
(1 point)
Responses
0.215
0.215
0.607
0.607
0.902
0.902
0.807
. Along one side of the lawn, there is a semicircular flower garden with a diameter of 10 ft
.
A square with a semicircle inside the square with the diameter of the circle on top of the left side of the square. The square’s side is labeled 20 feet and the semicircle’s diameter is labeled 10 feet.
A bird drops a seed randomly on the lawn. What is the probability that the seed lands outside the semicircular flower garden?
(1 point)
Responses
0.215
0.215
0.607
0.607
0.902
0.902
0.807
Answers
GPT-4o mini
To solve this problem, we need to calculate the area of the square lawn and the area of the semicircular flower garden. Then, we can find the area of the lawn that is outside the semicircular garden and calculate the probability that the seed lands outside.
1. **Area of the square lawn:**
\[
\text{Area}_{\text{square}} = \text{side}^2 = 20 \, \text{ft} \times 20 \, \text{ft} = 400 \, \text{ft}^2
\]
2. **Area of the semicircular flower garden:**
The radius of the semicircle is half of the diameter:
\[
\text{radius} = \frac{10 \, \text{ft}}{2} = 5 \, \text{ft}
\]
The area of the full circle would be:
\[
\text{Area}_{\text{circle}} = \pi \times \text{radius}^2 = \pi \times (5 \, \text{ft})^2 = 25\pi \, \text{ft}^2
\]
Since we only have a semicircle:
\[
\text{Area}_{\text{semicircle}} = \frac{1}{2} \times 25\pi \, \text{ft}^2 = \frac{25\pi}{2} \, \text{ft}^2 \approx 39.27 \, \text{ft}^2 \, (\text{using} \, \pi \approx 3.14)
\]
3. **Area of the lawn outside the semicircular garden:**
\[
\text{Area}_{\text{outside}} = \text{Area}_{\text{square}} - \text{Area}_{\text{semicircle}} = 400 \, \text{ft}^2 - \frac{25\pi}{2} \, \text{ft}^2
\]
\[
\text{Area}_{\text{outside}} \approx 400 \, \text{ft}^2 - 39.27 \, \text{ft}^2 \approx 360.73 \, \text{ft}^2
\]
4. **Probability that the seed lands outside the semicircular flower garden:**
\[
P(\text{outside}) = \frac{\text{Area}_{\text{outside}}}{\text{Area}_{\text{square}}} = \frac{360.73 \, \text{ft}^2}{400 \, \text{ft}^2} \approx 0.901825
\]
Therefore, rounding to three decimal places, the probability that the seed lands outside the semicircular garden is approximately 0.902.
The correct answer is:
**0.902**
1. **Area of the square lawn:**
\[
\text{Area}_{\text{square}} = \text{side}^2 = 20 \, \text{ft} \times 20 \, \text{ft} = 400 \, \text{ft}^2
\]
2. **Area of the semicircular flower garden:**
The radius of the semicircle is half of the diameter:
\[
\text{radius} = \frac{10 \, \text{ft}}{2} = 5 \, \text{ft}
\]
The area of the full circle would be:
\[
\text{Area}_{\text{circle}} = \pi \times \text{radius}^2 = \pi \times (5 \, \text{ft})^2 = 25\pi \, \text{ft}^2
\]
Since we only have a semicircle:
\[
\text{Area}_{\text{semicircle}} = \frac{1}{2} \times 25\pi \, \text{ft}^2 = \frac{25\pi}{2} \, \text{ft}^2 \approx 39.27 \, \text{ft}^2 \, (\text{using} \, \pi \approx 3.14)
\]
3. **Area of the lawn outside the semicircular garden:**
\[
\text{Area}_{\text{outside}} = \text{Area}_{\text{square}} - \text{Area}_{\text{semicircle}} = 400 \, \text{ft}^2 - \frac{25\pi}{2} \, \text{ft}^2
\]
\[
\text{Area}_{\text{outside}} \approx 400 \, \text{ft}^2 - 39.27 \, \text{ft}^2 \approx 360.73 \, \text{ft}^2
\]
4. **Probability that the seed lands outside the semicircular flower garden:**
\[
P(\text{outside}) = \frac{\text{Area}_{\text{outside}}}{\text{Area}_{\text{square}}} = \frac{360.73 \, \text{ft}^2}{400 \, \text{ft}^2} \approx 0.901825
\]
Therefore, rounding to three decimal places, the probability that the seed lands outside the semicircular garden is approximately 0.902.
The correct answer is:
**0.902**